Question 958546
{{{graph(300,300,-7,7,-2,2,-sin(x))}}}
A line tangent to the curve {{{y=-sin(x)}}} at point has a slope of {{{m=-cos(x)}}}.
Since it goes through point ({{{x}}},{{{y}}}) and the origin then the slope also equals.
{{{m=(y-0)/(x-0)=y/x=-sin(x)/x}}}
So then, 
{{{-sin(x)/x=-cos(x)}}}
{{{sin(x)-xcos(x)=0}}}
So to find the root, let
{{{f(x)=sin(x)-xcos(x)}}}
Then,
{{{df/dx=xsin(x)}}}
Now using Newton's method with a starting value of {{{x=4}}}
*[illustration new.JPG].
So then at {{{x=4.493409}}}, the slope is equal to,
{{{m=-cos(4.493409)}}}
{{{highlight(m=0.217234)}}}
{{{graph(300,300,-7,7,-2,2,-sin(x),0.217234x)}}}